The new version has been tested against Ghost 0.11.4 (i.e. that's the version I'm currently running), but should work on any version beyond Ghost 0.9.0 at the time of writing. It might also work on older or newer versions, but I make no guarantees. If you would like to know what has changed, you can check out the Ghost dev blog.
If you need help upgrading Ghost itself, just follow the instructions in their own Support page.
If you're not yet using StayPuft and would like to try it out, just take a look at the Installation section over at GitHub.
To update the theme, just follow the Installation instructions as normal. You may copy over your existing
partials/sidebar-external.hbs, but the
partials/copyright.hbs file is now deprecated.